Question

he complex formsof the sine and cosine functions are:ieexixix2sinand2cosixixeex, where eis the Euler e, and 1i.Differentiate the complex form of cosxto showthatxxdxdsin)(cos.[Hints:The Quotient Rule may be used, but it is not necessaryif you factor out a constant first.√−1=iis a constant. In fact, you can use it to multiply a fraction by iiif that helps...]

Answer 1

`\cos'(z) = -\sin(z)`

Explanation:

According to the information given by the problem

`\sin(z) = {\displaystyle (e^(iz) - e^(-iz) )/(2i) }`

`\cos(z) = {\displaystyle (e^(iz) + e^(-iz) )/(2) }`

Now, if you compute the derivative of
`\cos` you get that

`\cos'(z) = {\displaystyle ( ie^(iz)-i e^(iz) )/(2) } = {\displaystyle ( i ( e^(iz)- e^(iz) ))/(2) }\\\\= {\displaystyle ( i ( e^(iz)- e^(iz) ))/(2) } *(i)/(i) }\\\\= {\displaystyle - ( e^(iz)- e^(iz) )/(2i) } = -\sin(z)`